Find the point on the parabola y^2 = 4x that is closest to the point (2, 8).
Answer:
(4, 4)
Step-by-step explanation:
There are a couple of ways to go at this:
Write an expression for the distance from a point on the parabola to the given point, then differentiate that and set the derivative to zero.Find the equation of a normal line to the parabola that goes through the given point.1. The distance formula tells us for some point (x, y) on the parabola, the distance d satisfies ...
... d² = (x -2)² +(y -8)² . . . . . . . the y in this equation is a function of x
Differentiating with respect to x and setting dd/dx=0, we have ...
... 2d(dd/dx) = 0 = 2(x -2) +2(y -8)(dy/dx)
We can factor 2 from this to get
... 0 = x -2 +(y -8)(dy/dx)
Differentiating the parabola's equation, we find ...
... 2y(dy/dx) = 4
... dy/dx = 2/y
Substituting for x (=y²/4) and dy/dx into our derivative equation above, we get
... 0 = y²/4 -2 +(y -8)(2/y) = y²/4 -16/y
... 64 = y³ . . . . . . multiply by 4y, add 64
... 4 = y . . . . . . . . cube root
... y²/4 = 16/4 = x = 4
_____
2. The derivative above tells us the slope at point (x, y) on the parabola is ...
... dy/dx = 2/y
Then the slope of the normal line at that point is ...
... -1/(dy/dx) = -y/2
The normal line through the point (2, 8) will have equation (in point-slope form) ...
... y - 8 = (-y/2)(x -2)
Substituting for x using the equation of the parabola, we get
... y - 8 = (-y/2)(y²/4 -2)
Multiplying by 8 gives ...
... 8y -64 = -y³ +8y
... y³ = 64 . . . . subtract 8y, multiply by -1
... y = 4 . . . . . . cube root
... x = y²/4 = 4
The point on the parabola that is closest to the point (2, 8) is (4, 4).
As Saturn revolves around the sun, it travels at a speed of approximately 6 miles per second. Convert this speed to miles per minute. At this speed, how many miles will Saturn travel in 4 minutes? Do not round your answers.
Consider that lines u and v are parallel. Which equation models the relationship between the angles? What is the value of x? A) 12x - 4 = 10x + 10; x = 7 B) 12x - 4 + 10x + 10 = 180; x = 7.9 C) 12x - 4 = 10x - 10; x = -3 D) 12x + 4 = 10x + 10; x = 3
Answer:
A) 12x - 4 = 10x + 10; x = 7
Step-by-step explanation:
12x - 4 = 10x + 10; x = 7
Since lines u and v are parallel, the two angles are corresponding angles. Corresponding angles are equal to each other.
12x - 4 = 10x + 10
Solve for x:
2x = 14
x = 7
For a sample of n = 100 scores, x = 45 corresponds to z = 0.50 and x = 52 corresponds to z = +1.00. what are the values for the sample mean and standard deviation? m = 31 and s = 7 m = 31 and s = 14 m = 38 and s = 7 m = 38 and s = 14
We are given a fixed number of samples, n = 100.
We are given two conditions:
x = 45, z = 0.50
x = 52, z = 1.00
The relevant equation we can use here is:
z = (x – m) / s
where m is the mean and s is the std dev
So for the two conditions:
0.50 = (45 – m) / s --> eqtn 1
1.00 = (52 – m) / s --> eqtn 2
Rewriting eqtn 1 in terms of m:
0.5 s = 45 – m
m = 45 – 0.5 s --> eqtn 3
Rewriting eqtn 2 in terms of m:
1.00 s = 52 – m
m = 52 – 1.00 s --> eqtn 4
Equating eqtn 3 and 4:
45 – 0.5 s = 52 – 1.00 s
0.5 s = 7
s = 14
From eqtn 4:
m = 52 - 1.00 * 14
m = 38
Therefore answers are:
m = 38 and s = 14
find all solutions of the equation tan^5x-9tanx=0. the answer is Akipi. where k is any integer. the constant A=
To solve tan^5x - 9tanx = 0, we factor to get tanx(tan^4x - 9) = 0 leading to solutions where x = kπ and x = ±π/3 + kπ. The constant A in the solution Akiπ is determined to be ±π/3.
Explanation:To find all solutions to the equation tan^5x - 9tanx = 0, we can factor it as follows:
tanx(tan^4x - 9) = 0
This leads to two possible sets of solutions: tanx = 0 and tan^4x = 9.
For tanx = 0, x would be any integer multiple of π, i.e., x = kπ where k is an integer.
For tan^4x = 9, taking the fourth root gives us tanx = ±9√3. Since tangent is periodic with π, the solution would be of the form x = tan⁻¹(±9√3) + kπ, but since tan⁻¹(±9√3) simplifies to ±π/3, the solution can be written as x = ±π/3 + kπ.
However, if we are given that the solution is in the form Akiπ, we must determine the constant A. From the provided solutions, A must be a solution to tanx = 0 or tan⁻¹(±9√3), giving us A = 0 or ±π/3. Yet, since we cannot have a zero multiple of π (because that would give us a null solution), we dismiss A = 0 and take A from the non-zero solutions, so A = ±π/3.
use two unit multipliers to convert 56 centimeters to feet
Cell phones and surveys ii the survey by the national center for health statistics further found that 49% of adults ages 25–29 had only a cell phone and no landline. we randomly select four 25–29-year-olds:
a.what is the probability that all of these adults have a only a cell phone and no landline?
b.what is the probability that none of these adults have only a cell phone and no landline?
c.what is the probability that at least one of these adults has only a cell phone and no landline?
(a) The probability that all of these adults have a only a cell phone and no landline is [tex](0.49)^4[/tex]
(b) The probability that none of these adults have only a cell phone and no landline is [tex](0.51)^4[/tex].
(c) The probability that at least one of these adults has only a cell phone and no landline is [tex]1-(0.51)^4=0.9323[/tex].
According to the question, the survey by the national center for health statistics found that 49% of adults ages 25–29 had only a cell phone and no landline.
Probability that adults ages 25–29 had only a cell phone and no landline is [tex]P_1=0.49[/tex]
Probability that adults ages 25–29 have only a cell phone and no landline is
[tex]P_2=1-0.49\\P_2=0.51[/tex]
On selection of random 4 persons aged between 25–29-
(a) probability that all of these adults have a only a cell phone and no landline is [tex](0.49)^4[/tex]
(b) the probability that none of these adults have only a cell phone and no landline is [tex](0.51)^4[/tex].
(c) the probability that at least one of these adults has only a cell phone and no landline is [tex]1-(0.51)^4=0.9323[/tex].
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f the centripetal and thus frictional force between the tires and the roadbed of a car moving in a circular path were reduced, what would happen?
In Death Valley, California the highest ground temperature recorded was 94 degrees Celsius on July 15, 1972. In the formula C=5/9(F-32), C represents the temperature in degrees Celsius and F re[resents the temperature in degrees Fahrenheit. To the nearest degree, what is the highest ground temperature in Death Valley in Fahrenheit?
Answer:
The highest ground temperature in Death Valley is [tex]201^{\circ}F [/tex].
Step-by-step explanation:
We are given that in death valley , Callifornia the highest ground temperature recorded was [tex]94^{\circ}C[/tex]
We are given formula
[tex] C=\frac{5}{9}(F-32)[/tex]
Where C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit.
We have to find the highest ground temperature in Death Valley in Fahrenheit to the nearest degree
Using formula [tex] F=\frac{9}{5}C+32[/tex]
Substituting the value of temperature in Celsius
Then we get
[tex]F=\frac{9}{5}\times 94+32[/tex]
[tex]F=\frac{846}{5}+32[/tex]
[tex]F=169.2+32[/tex]
[tex]F=201.2^{\circ}F[/tex]
[tex]F=201^{\circ}F[/tex]
Hence, the highest ground temperature in Death Valley is [tex]201^{\circ}F [/tex].
example for empirical probability
Empirical probability is a form of probability that is based on the actual results of an experiment. It's computed by dividing the number of times an event occurs by the total number of observations or trials. Therefore, empirical probability varies depending on the outcomes of the experiment.
Explanation:The empirical probability, or experimental probability, comes from actual observations or experiments, unlike theoretical probability which is based purely on mathematical principles. An example of empirical probability can be found in a simple coin toss experiment. Let's say we toss a coin 100 times and heads comes up 55 times.
To calculate the empirical probability of getting heads, we would divide the number of times the event (getting heads) occurs by the total number of opportunities for the event to occur (the total number of tosses). In this case, the empirical probability is given by 55 (the occurrences of heads) divided by 100 (total coin tosses), giving us an empirical probability of 0.55 for getting heads.
Another example, in a traffic situation, would be to install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a certain period of time. This data would allow us to calculate the empirical probability of a car running the red light.
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Suppose you obtain a $1,300 T-note with a 9% annual rate, paid monthly, with maturity in 6 years. How much interest will be paid to you each month?
Answer:
simple interest = $9.75
Step-by-step explanation:
given data:
Principle = $1300
annual rate = 9% [tex]= \frac{9}{`12} = 0.75 [/tex]
time = 6 year =
we knwo that simple interest is given as
Simple interest [tex]= \frac{P\times R\times T}{100}[/tex]
FOR ABOVE QUESTION
Time is 1 month
simple interest [tex]= \frac{1300\times 0.75 \times 1}{100}[/tex]
simple interest = $9.75
what is 0.04 as a standard form
Answer:
4.0 * 10^-2
Step-by-step explanation:
Assuming that Standard Form is using Scientific Notation, then you would move the decimal until it is directly after the 4. Then, you would multiply that by 10 raised to the negative exponent of how many spaces that you had to move the decimal. In this case, you moved it two places to the right (+), SO THE EXPONENT IS NEGATIVE!
0.04 = 4.0 * 10^-2
Eben rolls two standard number cubes 36 times. Predict how many times he will roll a sum of 4.
What happens to the area of a circle when the radius is tripled?
A 21-inch piece of steel is cut into three pieces so that the second piece is twice as long as the first piece, and the third piece is three inches more than six times the length of the first piece. Find the lengths of the pieces
(h) when is the particle speeding up? (enter your answer using interval notation.) (2,4)∪(6,8) incorrect: your answer is incorrect. f(t) = cos(πt/4)
Joanna wants to buy a car. Her parents loan her 5,000 for 5 years at 5% simple interest. How much will Joanna pay in imterest?
The Leukemia and Lymphoma Society sponsors a 5K race to raise money. It receives $55 per race entry and $10,000 in donations, but it must spend $15 per race entry to cover the cost of the race. Write and solve an inequality to determine the number of race entries the charity needs to raise at least $55,000.
In 2018, a nation’s population was 10 million. Its nominal GDP was $40 billion, and its price index was 100. In 2019, its population had increased to 12 million, its nominal GDP had risen to $57.6 billion, and its price index had increased to 120. What was this nation’s economic growth rate during the year?
The nominal GDP in base year 2014 was $40 billion. The nominal GDP in year 2015 with price index 120 was $57.6 billion. The real GDP in 2015 can be calculated as follows :
GDP (real) = GDP (nominal) / price index * 100
GDP (real) = 57.6 / 120 * 100
GDP (real) = $48 billion
The growth rate in real GDP from 2014 to 2015 is 1.2%.
Growth rate = 48 * (100/40) = 1.2%
Therefore the growth rate is 12%
The nation's real GDP increased from $40 billion in 2018 to $48 billion in 2019, resulting in an economic growth rate of 20% for that year.
To calculate the economic growth rate of a nation, we need to look at the increase in its real GDP. Real GDP is calculated by dividing the nominal GDP by the GDP deflator and then multiplying by 100. The GDP deflator is like a price index that reflects the level of prices of all new, domestically produced, final goods and services in an economy.
For the nation in question:
2018 Real GDP = (Nominal GDP / Price Index)To find the economic growth rate, we subtract the previous year's real GDP from the current year's real GDP, divide by the previous year's real GDP, and then multiply by 100 to get a percentage:
Economic Growth Rate = [(2019 Real GDP - 2018 Real GDP) / 2018 Real GDP]
100
Economic Growth Rate = [($48 billion - $40 billion) / $40 billion]
100 = (8 / 40)
100 = 20%
The nation's economic growth rate during the year was 20%.
Greta completed a mile race in 5 minutes . inez ran a mile in which each quarter mile split was 1 min 20 seconds which of the two girls had the fastest time? How much faster?
which equation best represents the model?
A publisher displays its latest magazine cover on its website.
The publisher scales up the front cover of the magazine using a scale of 6 centimeters to 1 inch. The length of the scale drawing is 48 centimeters, and its width is 66 centimeters.
The length of the actual magazine cover is inches.
The width of the actual magazine cover is inches.
The scale drawing is too big to view on a computer screen without scrolling.
the publisher uses a new scale of 4 centimeters to 1 inch.
The length of the new scale drawing is centimeters.
The width of the new scale drawing is centimeters.
Answer: 8 inches.11 inches.32 centimeters. 44 centimetres.
Step-by-step explanation:
The graph shows a line and two similar triangles.
Answer:
Option (a) is correct.
The equation of the line is expressed using expression [tex]\frac{y}{x}=\frac{1}{4}[/tex]
Step-by-step explanation:
Given : The graph shows a line and two similar triangles.
We have to find the expression that finds the equation of line.
Since, given two triangles are similar.
So, Δ ABC ≅ Δ ADE
Thus, There corresponding sides are in same proportion.
[tex]\frac{AC}{AD}=\frac{CB}{DE}= \frac{AB}{BE}[/tex]
Substitute, we get,
[tex]\frac{1}{y}=\frac{4}{x}= \frac{AB}{BE}[/tex]
Rearrange, we have,
[tex]\frac{1}{4}=\frac{y}{x}= \frac{AB}{BE}[/tex]
Also, finding slope of line AE,
Coordinate of B is (4,1) and Coordinate of A is (0,0)
Th equation of line is y = mx + c
Where, m is slope and c is x intercept
Since, [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{1-0}{4-0}[/tex]
Simplify, we have,
[tex]m=\frac{1}{4}[/tex]
And c = 0
Thus, equation of line is[tex]y=\frac{1}{4}x[/tex]
We re-writing we get,
[tex]\frac{y}{x}=\frac{1}{4}[/tex]
Thus, The equation of the line is expressed using expression [tex]\frac{y}{x}=\frac{1}{4}[/tex]
Given the following geometric sequence, find the common ratio: {225, 45, 9, ...}.
Answer: The required common ratio for the given geometric sequence is [tex]\dfrac{1}{5}.[/tex]
Step-by-step explanation: We are given to find the common ratio for the following geometric sequence :
225, 45, 9, . . .
We know that
in a geometric sequence, the ratio of any term with the preceding term is the common ratio of the sequence.
For the given geometric sequence, we have
a(1) = 225, a(2) = 45, a(3) = 9, etc.
So, the common ratio (r) is given by
[tex]r=\dfrac{a(2)}{a(1)}=\dfrac{a(3)}{a(2)}=~~.~~.~~.~~.[/tex]
We have
[tex]\dfrac{a(2)}{a(1)}=\dfrac{45}{225}=\dfrac{1}{5},\\\\\\\dfrac{a(3)}{a(2)}=\dfrac{9}{45}=\dfrac{1}{5},~etc.[/tex]
Therefore, we get
[tex]r=\dfrac{1}{5}.[/tex]
Thus, the required common ratio for the given geometric sequence is [tex]\dfrac{1}{5}.[/tex]
Two arrows are launched at the same time with the same speed. arrow a at an angle greater than 45 degrees, and arrow b at an angle less than 45 degrees. both land at the same spot on the ground. which arrow arrives first?
The base and height of Triangle A are half the base and the height of Triangle B. How many times greater is the area of Triangle B?
Maria incorrectly placed the decimal point when she wrote 0.65 inch fo the width of her computer. what is the correct decimal number for the width?
Answer:
It should be placed after 6 it means 6.5inch
Step-by-step explanation:
Maria should placed the decimal point after 6 it means 6.5 inch. Because the order of the width of her computer should be in between 6 inch to 9 inches.
Width of a computer cannot be 0.65 inches because it will be too short and we cannot called it computer on the other hand if she put the decimal after 5 then the width of her computer will be 65 inches which is not normal.
Dominic has $15 for dinner. His meal costs $13.90. He wants to leave an 18% tip. Does he have enough money? Explain your reasoning
um plz help On a soccer team, 11 out of 17 players surveyed say they had two or more siblings. The league has 850 players. Which is the best prediction of the number of players in the league that have two or more siblings?
Verify stokes' theorem for the helicoid ψ(r,θ)=⟨rcosθ,rsinθ,θ⟩ where (r,θ) lies in the rectangle [0,1]×[0,π/2], and f is the vector field f=⟨6z,8x,8y⟩. first, compute the surface integral: ∬m(∇×f)⋅ds=∫ba∫dcf(r,θ)drdθ, where a= , b= , c= , d= , and f(r,θ)= (use "t" for theta). finally, the value of the surface integral is . next compute the line integral on that part of the boundary from (1,0,0) to (0,1,π/2). ∫cf⋅dr=∫bag(θ)dθ, where a= , b= , and g(θ)=
Stokes' theorem, which relates a surface integral of a curl of a vector field over a surface to a line integral of the vector field over the boundary of the surface, can be applied to verify a surface described by a helicoid and a given vector field, through calculation of the surface and line integrals, even when specific function values are not provided.
Explanation:Stokes' theorem relates a surface integral of a curl of a vector field over a surface Ψ to a line integral of the vector field over the boundary ∂Ψ of the surface. Given a helicoid ψ(r,θ)=⟨rcosθ,rsinθ,θ⟩ where (r,θ) lie in the rectangle [0,1]×[0,π/2], and f is the vector field f=⟨6z,8x,8y⟩, Stokes' theorem can be applied to verify the vectro field over the given area.
The process involves two primary steps: computation of the surface integral, and computation of the line integral.
Step 1: Compute the surface integral ∬m(∇×f)⋅ds=∫ba∫dcf(r,θ)drdθ. However, because the specific values for a, b, c and d, and the function f(r,θ) are not defined in this question, the exact calculation can't be provided.
Step 2: Compute the line integral ∫cf⋅dr=∫bag(θ)dθ, on the boundary from (1,0,0) to (0,1,π/2). Again, specific values for a, b and the function g(θ) are not provided.
According to Stokes' theorem, the two results should be equal.
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